sensors

Communication Towards a Highly Sensitive Piezoelectric Nano-Mass Detection—A Model-Based Concept Study

Jens Twiefel 1,* , Anatoly Glukhovkoy 2, Sascha de Wall 2, Marc Christopher Wurz 2, Merle Sehlmeyer 3, Moritz Hitzemann 3 and Stefan Zimmermann 3

1 Institute of Dynamics and Vibration Research, Leibniz Universität Hannover, An der Universität 1 Geb. 8142, 30823 Grabsen, Germany 2 Institute of Micro Production Technology, Leibniz Universität Hannover, An der Universität 2, 30823 Grabsen, Germany; [email protected] (A.G.); [email protected] (S.d.W.); [email protected] (M.C.W.) 3 Institute of and Measurement Technology, Leibniz Universität Hannover, Appelstr. 9A, 30167 Hannover, Germany; [email protected] (M.S.); [email protected] (M.H.); [email protected] (S.Z.) * Correspondence: [email protected]; Tel.: +49-511-762-4167

Abstract: The detection of exceedingly small masses still presents a large challenge, and even though very high sensitivities have been archived, the fabrication of those setups is still difficult. In this paper, a novel approach for a co-resonant mass detector is theoretically presented, where simple fabrication is addressed in this early concept . To simplify the setup, longitudinal and bending vibrations were combined for the first time. The direct integration of an aluminum nitride (AlN)   piezoelectric element for simultaneous excitation and sensing further simplified the setup. The feasibility of this concept is shown by a model-based approach, and the underlying parameter Citation: Twiefel, J.; Glukhovkoy, A.; de Wall, S.; Wurz, M.C.; Sehlmeyer, dependencies are presented with an equivalent model. To include the geometrical and material M.; Hitzemann, M.; Zimmermann, S. aspects, a finite element model that supports the concept as a very promising approach for future Towards a Highly Sensitive nano-mass detectors is established. Piezoelectric Nano-Mass Detection— A Model-Based Concept Study. Sensors Keywords: piezoelectric sensors; nano-mass detection; inertial balance; systems; nano/micro- 2021, 21, 2533. https://doi.org/ electro-mechanical-system; N/MEMS; co-resonance 10.3390/s21072533

Academic Editor: Christopher Oshman 1. Introduction The detection of weight is an important measurement task that has been performed Received: 2 March 2021 for thousands of years. Over time, precision and repeatability has dramatically improved. Accepted: 2 April 2021 Published: 4 April 2021 However, the measurement of exceedingly small masses—of the order of the atomic mass unit Dalton—is still a great challenge, even today. An important method for the detection

Publisher’s Note: MDPI stays neutral of the smallest masses is the use of resonance modes, wherein the resonance is with regard to jurisdictional claims in detuned by the mass to be detected. This method was first described in [1] in 1959 and has −10 published maps and institutional affil- been since then cited more than 9000 times. At that time, an accuracy of 10 g was already iations. archived with eigenfrequencies in the order of 10 MHz. Such an inertial balance utilizes the piezoelectric effect of the vibrating element, which is exited at a fundamental resonance via a closed loop control (e.g., [2]). This can be realized by either a phase-locked-loop or a self-oscillating circuitry. The frequency dependence of the resonance on the analyte mass is due to the ratio of analyte mass to the effective mass of the resonator; the highest sensitivity Copyright: © 2021 by the authors. Licensee MDPI, Basel, Switzerland. is obtained when both masses are similar in weight. Consequently, research in this field This article is an open access article has focused on even smaller and lighter resonator structures while aiming for better mass distributed under the terms and resolutions. The smallest known resonators are carbon nanotubes, with flexural mode conditions of the Creative Commons in the GHz range that allow for yoctogram mass resolutions [3]. However, the Attribution (CC BY) license (https:// constantly decreasing sizes bring two major challenges: first, the fabrication gets more creativecommons.org/licenses/by/ complicated and therefore more expensive, making reproducibility difficult; second, the 4.0/). vibration measurement increases in complexity with decreasing structure size.

Sensors 2021, 21, 2533. https://doi.org/10.3390/s21072533 https://www.mdpi.com/journal/sensors Sensors 2021, 21, 2533 2 of 15

In recent years, many research groups have worked on improving inertial balances, as listed in Table1. Various concepts of actuation and mass sensing have been evaluated including inductive, electrostatic, piezoelectric, and piezoresistive methods [4]. The max- imum obtained resolution was 10−24 g [5,6] at temperature levels from 5 to 300 K and ambient pressures of 10−10 to 10−5 Torr. Selected excitation and measurement methods are described below as examples. In the inductive method, a nano-resonator is placed in a magnetic field and a broadband alter- nating current is passed through the resonator so that the Lorentz force excites . The oscillation, in turn, induces a voltage that is amplified and measured to record the oscillation. A disadvantage, however, is the superposition of the measurement signal and the excitation. With the aid of special circuit variants, the feedback effect of the exciting signal on the measurement signal can be reduced [7,8].

Table 1. Selected publications on resonant nano-mass detection systems.

Beam Measurement Material Resolution [g] ω [MHz] Co-Resonant Excitation Support 0 Principle [9] Silicon single-sided 5.5 × 10−15 1...10 - photothermal interferometer [10] Silicon double-sided n.a. 1000 - capacitive capacitive [11] SiC double-sided 2.53 × 10−18 32.8 - capacitive capacitive [12] CNT single-sided 1.3 × 10−22 328.5 - capacitive capacitive [13] CNT double-sided 2.5 × 10−20 125 - capacitive reflection [14] SiC double-sided n.a. 428 - capacitive capacitive [5] CNT double-sided 1.7 × 10−24 2000 - capacitive reflection [6] CNT single-sided 1.7 × 10−24 12 × 104 - capacitive reflection [15] Graphene double-sided 1.41 × 10−21 1.1 + piezoelectric piezoresistive [16] Silicon double-sided 1.7 × 10−21 20...120 + capacitive piezoresistive [17] Nano-crystalline single-sided 10−18 12 × 103 + capacitive reflection [18] Silicon single-sided 10−12 1.1 + piezoelectric piezoelectric

In the electrostatic method, a nano-resonator is excited by an electrostatic field. A voltage between the nanostructure and electrodes, which are arranged in parallel, usually generates the field. In the case, either a broadband source or a source with an adjustable frequency can serve as the voltage source, whereby the frequency must be monitored for the latter, e.g., via a phase-locked loop. The resonant frequency is detected by measuring the change in capacitance between the nano-resonator and the counter electrode. The capacitance change can be measured by measuring the frequency of the displacement current at the capacitor with a transimpedance amplifier [4,7,19]. Self-oscillating active electrical circuits that integrate the nano-resonator circuitry are also used. Another method for determining the resonant frequency is based on the reflection measurement at an integrated transistor, which consists of a semiconducting nano-resonator that is mechanically clamped and electrically contacted on both sides with a counter electrode as gate. The reflection of an alternating signal at the electrical RLC resonant circuit with the integrated nano-transducer is then measured. Particle adsorption at the nano-resonator leads to a change in resistance and capacitance at a constant voltage between the gate and the oscillator and thus to a change in the reflection. However, this method requires a complex design of the nano-resonator as a transistor [20,21]. Other methods for measuring are based on optical interferometers in which an incident light beam is reflected by the nano-oscillator and superimposed with a reference beam. Based on the time-varying interference pattern, the vibration can be analyzed. The excitation of the nano-resonator can also be done photothermally with a laser by heating and expanding the nanostructure [9,22]. Piezoelectric methods, e.g., see [18], are based on the mechanical excitation of the nano-resonator via the deformation of piezoelectric nanostructures when a voltage is applied. Similar to electrostatic methods, the electrical voltage for vibration excitation can Sensors 2021, 21, x FOR PEER REVIEW 3 of 15

reference beam. Based on the time-varying interference pattern, the vibration can be ana- lyzed. The excitation of the nano-resonator can also be done photothermally with a laser by heating and expanding the nanostructure [9,22]. Sensors 2021, 21, 2533 Piezoelectric methods, e.g., see [18], are based on the mechanical excitation 3of of the 15 nano-resonator via the deformation of piezoelectric nanostructures when a voltage is ap- plied. Similar to electrostatic methods, the electrical voltage for vibration excitation can be begenerated generated via via a self-oscillating a self-oscillating active active electrical electrical circuit, circuit, e.g., e.g., see see [23]. [23 The]. The piezoelectric piezoelectric ap- approachproach of ofvibration vibration excitation excitation and and measuring measuring the the resonant resonant frequency frequency was was applied applied in this in thisstudy. study. AA fundamentalfundamental remaining remaining challenge challenge is designing is designing and, and, in particular, in particular, fabricating fabricating a highly a accuratehighly accurate and reproducible and reproducible mass detection mass detection systems thatsystems would that allow would for theallow detection for the ofdetec- the smallesttion of the masses smallest in a masses simple in way a simple with minimal way with effort, minimal so that, effort, for so example, that, for large example, arrays large or singlearrays sensors or single for sensors process for or process environmental or environmental monitoring monitoring become possible. become possible. InIn thisthis preliminarypreliminary conceptconcept study,study, we we focused focused on on co-resonance co-resonance systems, systems, utilizing utilizing the the advantageadvantage thatthat thethe frequencyfrequency detectiondetection cancan bebe relocatedrelocated fromfromthe thetiny tiny nano-resonatornano-resonator to to a a secondsecond somewhat-largesomewhat-large resonator,resonator, which which is is the the host host platform platform forfor thethe nano-resonator.nano-resonator. This This conceptconcept isis extended extended using using a piezoelectrica piezoelectric element elem forent thefor largerthe larger resonator. resonator. If well-designed, If well-de- thissigned, would this allow would for allow the use for ofthe the use piezoelectric of the piezoelectric effect for effect both for the both actuation the actuation of the nano-of the beamnano-beam and the and detection the detection of its resonance of its resonance frequency. frequency. The fabrication The fabric of nano-beamsation of nano-beams that are attachedthat are attached to the tips toof the larger tips of beams—which larger beams—which is the usual is the setup usual of co-resonantsetup of co-resonant systems—is sys- rathertems—is complicated. rather complicated. We therefore We proposetherefore a pr differentopose a approach. different approach. Here, the largerHere, resonatorthe larger wasresonator designed was in designed a longitudinal in a longitudinal fashion. In fashion. addition In to addition the ease to of the fabrication, ease of fabrication, this leads tothis an leads even to vibration an even distribution.vibration distribution. On a respective On a respective surface, thesurface, sensitivity the sensitivity to additional to ad- massesditional is masses approximately is approximately equal across equal the across entire the surface. entire Since surface. the Si surfacence the is surface the free is end, the thefree vibration end, the amplitudevibration is the maximum is the ma (anti-nodal-plane)ximum (anti-nodal-plane) of the longitudinal of the longitudinal vibration. Thisvibration. gives aThis certain gives degree a certain of freedom degree of for freedom the setup for of the the setup nano-resonator. of the nano-resonator. A schematic A sketchschematic of the sketch novel of concept the novel is givenconcept in Figureis given1. in Figure 1.

FigureFigure 1.1. NovelNovel setupsetup forfor aa piezoelectricpiezoelectric co-resonantco-resonant mass-detector.mass-detector. The piezoelectric element was designeddesigned asas aa longitudinallongitudinal resonator,resonator, and and the the nano-beam nano-beam was was operated operated in in a a bending bending mode. mode.

InIn thisthis study,study, we we investigated investigated the the feasibility feasibility of of this this novel novel concept. concept. This This included included study- stud- ingying relevant relevant design design parameters. parameters. For For this this purpose, purpose, three three model model approaches approaches are are presented presented inin thethe MaterialsMaterials andand Method section and and discus discussedsed in in the the Results Results section. section. The The goal goal was was to toreveal reveal parameter parameter ranges ranges that that allow allow for for a a practical practical realization realization of of the proposedproposed concept.concept. Therefore,Therefore, the the sensitivity sensitivity of of the the vibrationvibration modes, modes, their their coupling,coupling, dependence dependence on on mass, mass, and and stiffnessstiffness propertiesproperties werewere investigatedinvestigated byby utilizingutilizing an an equivalent equivalent model. model. TheThe piezoelectricpiezoelectric effecteffect alsoalso neededneeded toto bebe considered in order to evaluate the the sensing sensing performance. performance. Using Using a a finite element (FE) model, the geometry of the proposed structure was designed and specified. The resulting model-based findings were used to evaluate the characteristic of the novel setup.

2. Materials and Methods The first step was the selection of useful eigenmodes. For this purpose, we investigated their frequency sensitivity of the bending and longitudinal modes. Those studies used a point mass representing the analyte mass, located in the anti-node of the vibration. Sensors 2021, 21, 2533 4 of 15

Furthermore, an equivalent model for the investigation of co-resonant vibrations was developed. This was used to compute the major parameter affecting the vibration. Finally, the parameter of the later finite element model was presented, and this model was then utilized to show the feasibility of the novel concept.

2.1. Sensitivity of Relevant Mode Shapes To investigate the influence of the analyte mass on the resonance frequency for the interesting mode shapes, we used the well-known, analytically-derived formulas given in many text-books; here, we used the reference book [24]. For the beam, three configurations were investigated: clamped-free (cf), with the extra mass at the free end, and clamped– clamped (cc) and pinned–pinned (pp), both with the extra mass at the center position. The resonance f , where the index Bxx indicates the bending vibration and the respective boundary conditions, are as follows. s 1 3EI = fBcf 3 , (1) 2 π l (mn + 0.24 m) s 4 3EI = fBcc 3 , (2) π l (mn + 0.37 m) s 4 3EI = fBpp 3 , (3) π l (mn + 0.49 m) where E is the modulus of elasticity, l is the free length of the beam, I is the area moment of inertia, m is the mass of the uniform beam, and mn is the analyte mass. In addition to the bending, we also considered the longitudinal vibration. The boundary conditions of interest were: clamped-free (cf) and free–free (ff), both with an extra mass at one free side. The resonance frequencies were mathematically determined as follows: s λ E m f = cf with cot(λ ) = n λ , (4) Lcf 2 π l $ cf m cf

s λ E m f = ff with tan(λ ) = − n λ , (5) Lff 2 π l $ ff m ff

where $ is the density and λcf and λff are the dimensionless frequency parameters that can be found by solving the given transcendental equation, where we are interested in the first solution for the fundamental eigenfrequency. In case of mn = 0, the parameters are: λcf = π/2 and λff = π. For the investigation of the frequency shift dependent on the analyte mass, we com- puted the ratio of the loaded and unloaded resonance frequencies (marked with the index 0) and subtracted 1 to obtain the relative frequency change. Additionally, we introduced the mass ratio γ = mn/m and obtained the relative frequency shift β for the five cases: s fBcf 1 βBcf = − 1 = γ − 1 (6) fBcf,0 0.24 + 1 s fBcc 1 βBcc = − 1 = γ − 1 (7) fBcc,0 0.37 + 1 s fBcf 1 βBcf = − 1 = γ − 1 (8) fBcf,0 0.49 + 1 Sensors 2021, 21, x FOR PEER REVIEW 5 of 15

1 = −1= −1 (7) , +1 0.37

1 = −1= −1 (8) , +1 0.49

Sensors 2021, 21, 2533 = −1= −1 with cot()=, 5 of 15(9) ,

= −1= −1 with tan() =−, (10) , fLcf 2 λcf βLcf = − 1 = − 1 with cot(λcf) = γλcf, (9) In the special case thatfLcf,0 the mass ratiosπ were much smaller than one, all frequency ratios were very similar and was in the same order of magnitude as the mass ratio. The f λ results for the bendingβ =frequenciesLff − 1 =are summariff − 1 withzed tan in( Figureλ ) = −2. γλFor ,small mass ratios, (10) the Lff f π ff ff function was linear, and forLff,0 larger mass ratios (≈>0.05), the approximation of the point massIn was the insufficient. special case that the mass ratios were much smaller than one, all frequency ratiosAs were an veryexample, similar the and followingβ was inapproximation the same order could of magnitude be made using as the the mass formulas. ratio. The If an resultsanalyte for mass the bendingis a billion-times frequencies smaller are summarized than the mass in Figureof a resonator,2. For small the frequency mass ratios, resolu- the functiontion of the was measurement linear, and for electronics larger mass has ratios to be (≈ betterγ > 0.05 than), thea billionth approximation of the resonance of the point fre- massquency. was insufficient.

FigureFigure 2. 2.Relative Relative frequency frequency changes changes (− (−β) of) of bending bending modes modes over over mass mass ratio ratio (γ ().).

AsThus, an example, the mass the of followingthe nano-resonator approximation is crucial could for be madethe mass using resolution, the formulas. especially If an analytewhen considering mass is a billion-times the required smaller frequency than the resolution mass of aof resonator, the measurement the frequency electronics. resolution If, at ofa theresonance measurement frequency electronics of 1 GHz, has 1 to mHz be better (the resolution than a billionth of a good of the impedance resonance analyzer) frequency. is detectable,Thus, the the mass detection of the of nano-resonator a mass 1012 times is crucial smaller for than the the mass resonator resolution, is theoretically especially whenpossible. considering In other the words, required to detect frequency a yoctog resolutionram, the of themass measurement of the nano-resonator electronics. must If, at abe resonancesmaller than frequency a picogram. of 1 GHz, However, 1 mHz many (the resolutiondisturbing ofinfluences a good impedance (temperature, analyzer) pressure, is detectable,etc.) can negatively the detection affect of the a mass resolution. 1012 times smaller than the resonator is theoretically possible. In other words, to detect a yoctogram, the mass of the nano-resonator must be smaller2.2. Equivalent than a picogram.Model for A However, Piezoelectric many Co-Resonant disturbing Vibration influences System (temperature, pressure, etc.) canThe negatively direct excitation affect the and resolution. measurement of a nano-resonator is challenging; therefore, we used and extended the co-resonance concept. The main idea was to couple a smaller 2.2. Equivalent Model for A Piezoelectric Co-Resonant Vibration System and a larger resonator in such a way that a change of the vibration properties of the smaller resonatorThe direct could excitation be measured and measurement by the vibration of a pr nano-resonatoroperties of the islarger challenging; resonator. therefore, For this wepurpose, used and two extended vibrational the structures co-resonance were concept. selected The (typically main idea two was bending to couple beams) a smaller and in- anddependently a larger resonator tuned to in the such same a way resonance that a change frequency. of the Then, vibration the propertiessmaller beam of the was smaller placed resonatoron the larger could beam be measured at a position by thewhere vibration the selected properties eigenmode of the largerof the larger resonator. beam For had this the purpose, two vibrational structures were selected (typically two bending beams) and highest amplitude (anti-node). At this position, the impact of the smaller beam on the independently tuned to the same resonance frequency. Then, the smaller beam was placed larger beam was maximized. Due to the physical combination of the two structures, a new on the larger beam at a position where the selected eigenmode of the larger beam had the highest amplitude (anti-node). At this position, the impact of the smaller beam on the larger beam was maximized. Due to the physical combination of the two structures, a new system was created with new properties, with respect to eigenfrequencies and mode shapes. For analytical purposes, we applied a simplified model based on a system with ideal lumped parameters; two degrees of freedom; and discrete stiffnesses, damper, and masses. The piezoelectric effect—which is used for actuation and sensing—can be included through the second electro-mechanical analogy, e.g., see [25]. By utilizing a parameter comparison of the equations for electrical and mechanical discrete components, the analogy was traditionally derived. In the herein-used second analogy, displacement is equivalent to electric charge and force is equivalent to current. An ideal transmission element such as Sensors 2021, 21, x FOR PEER REVIEW 6 of 15

system was created with new properties, with respect to eigenfrequencies and mode shapes. For analytical purposes, we applied a simplified model based on a system with ideal lumped parameters; two degrees of freedom; and discrete stiffnesses, damper, and masses. The piezoelectric effect—which is used for actuation and sensing—can be in-

Sensors 2021, 21, 2533 cluded through the second electro-mechanical analogy, e.g., see [25]. By utilizing a param-6 of 15 eter comparison of the equations for electrical and mechanical discrete components, the analogy was traditionally derived. In the herein-used second analogy, displacement is equivalent to electric charge and force is equivalent to current. An ideal transmission ele- ament lever such can transferas a lever energy can transfer from oneenergy domain from to one the domain other. However,to the other. the However, lever parameter the lever consequentlyparameter consequently has a unit. The has modela unit. isTh depictede model inis Figuredepicted3. in Figure 3.

Figure 3. Simplified equivalent model for the co-resonant mass detector. Figure 3. Simplified equivalent model for the co-resonant mass detector.

TheThe model model has has three three coordinates: coordinates: thethe absoluteabsolute positionposition yof of the the lager lager mass massM ,, the the positionpositionx of of the the smaller smaller mass massm with with respect respect to toy , and, and the the electric electric charge chargeQ . The. The larger larger mass is inertially connected though the stiffness and the damper . The stiffness mass M is inertially connected though the stiffness kM and the damper dM. The stiffness k and the damper connects and . The analyte mass is adding to . The lever and the damper d connects M and m. The analyte mass is mn adding to m. The lever with thewith unitized the unitized factor factorα couples α couples the mechanic the mechanic and the and electric the electric domain. domain. Thecapacity The capacity of the of p piezoelectricthe piezoelectric element element is Cp is, and C , theand system the system is driven is driven by the by voltage the voltageU. Utilizing U. Utilizing complex com- amplitudesplex amplitudesx(t) = ()Re =xˆe Rejϕe(jΩet =e Re) =RexˆejΩteand theand imaginary the imaginary unit junit and jΩ andas angularΩ as an- excitationgular excitation frequency, frequency, for the equationsfor the equations of motion of followsmotion forfollows the harmonic for the harmonic excitation: excita- tion: −(m + m )Ω2 + jΩd + kxˆ = (m + m )Ω2yˆ, (−( +n ) +j+) =(+n ) 2  , −MΩ + jΩdM + kM yˆ − (jΩd + k)xˆ = αUˆ , and (11) (− +j1 Q+ˆ − )αyˆ−=(jUˆ . + ) =, and Cp Cp (11) − =. m = Additionally, we defined the two decoupled eigenfrequencies with n 0: r Additionally, we defined the two decoupledk eigenfrequencies with =0: ω = (12) 1 m r= (12) k ω = M (13) 2 M For further analysis, transfer-functions were also needed; here, we used the current amplitude iˆ = jΩQˆ : = (13)  iˆ j Ω α2 −mΩ2 + jΩd + k = = For further analysis, transfer-functions were also needed; here, we used+ the current GiU 2 3 4 jΩCp (14) Uˆ kkM + j(dMk + dkamplitudeM)Ω − (dd ̂M=j+ (k:+ kM)m + kM)Ω − j(dMm + d(m + M))Ω + mMΩ

  ̂ α − 2 + + yˆ = = mΩ jΩd k +j (14) = = ()(()) ( ()) GyU 2 3 4 (15) Uˆ kkM + j(dMk + dkM)Ω − (ddM + (k + kM)m + kM)Ω − j(dMm + d(m + M))Ω + mMΩ xˆ αmΩ2 = = GxU 2 3 4 (16) Uˆ kkM + j(dMk + dkM)Ω − (ddM + (k + kM)m + kM)Ω − j(dMm + d(m + M))Ω + mMΩ )

2.3. Finite Element Simulation To confirm that the novel idea of combining a longitudinal resonator with a bending resonator was promising approach for future nano-mass detectors, a 3D-FE model was built in ANSYS for first investigations. In this step, the substrate was included as an elastic foundation. The material parameters used for aluminum nitride (including the Sensors 2021, 21, 2533 7 of 15

piezoelectricity) and silicone dioxide stemmed from [26] and are given in Table2. For the analysis, we used modal and harmonic simulations.

Table 2. Used material parameters. AlN: aluminum nitride, Au: gold, SiO2: silicon dioxide.

Property Unit AlN Au SiO2 Density kg/m3 3512 19,320 2200 Poisson ratio - 0.3 0.42 0.17 Stiffness coefficient c11 GPa 345 Stiffness coefficient c12 GPa 125 Stiffness coefficient c13 GPa 120 Stiffness coefficient c33 /Elastic modulus GPa 395 79 70 Stiffness coefficient c44 GPa 118 Stiffness coefficient c66 GPa 110 2 Piezoelectric constant e31 C/m −0.58 2 Piezoelectric constant e33 C/m 1.55 2 Piezoelectric constant e15 C/m −0.48 Relative permittivity ε/ε0 - 11

3. Results and Discussion This section is divided in two subsections. First, the equivalent model is used to investigate the parameter impacts on the effective coupling between the two resonators. These results are then discussed in conjunction with theoretical sensitivity. Second, the feasibility of the concept, as presented in Figure1, is studied.

3.1. Evaluation of Equivalent Model For the investigation of the equivalent model, we concentrated on the interaction of the two modes, the frequency responses, and the sensitivity to the analyte mass.

3.1.1. Coupling of the Co-Resonant Masses

It is clear that ω1 = ω2 leads to the best interaction between the modes; this is well- known for a mechanically-tuned mass damper. For our purposes, the coupling had to be sufficiently large so that a resonance frequency shift of the nano-resonator changed the resonance frequency of the larger resonator. In our case, the mass M was always larger than m, and the coupling decreased with a smaller mass ratio κ = m/M. This was clear for a very large M, where any motion of m was negligible for the behavior of M. In the case that m and M were similar in weight, an interaction existed. One possibility to show this interaction was to investigate the system’s eigenfrequencies. For this, we focused on the undamped system. The mass ratio κ = m/M was used to scale the stiffness kM to ensure that both uncoupled eigenfrequencies remained equal ω1 = ω2 in the first place. For the investigation of a mistuning of the two eigenfrequencies, the factor Λ was introduced so that ω02 = Λω1. For simplicity, we used the coordinate xA for the mass m with respect to the reference frame. With Q 6= 0 and U = 0 as the boundary condition for the electric port and mn = 0 the mass and stiffness matrix, the following equation applies:

 m 0   k −k   x  M = , K = , x = A . (17) 0 m/κ −k Λ2 k/κ y

2 k The coupled system has the two eigenfrequencies with ω1 = m : r ! 1  2 ω2 = ω2 1 + Λ2 − 4κ + Λ2 − 1 (18) L 2 1 Sensors 2021, 21, x FOR PEER REVIEW 8 of 15

= in the first place. For the investigation of a mistuning of the two eigenfrequen- cies, the factor Λ was introduced so that ′ =Λ. For simplicity, we used the coordi- nate for the mass with respect to the reference frame. With 0 and =0 as the boundary condition for the electric port and =0 the mass and stiffness matrix, the following equation applies: 0 − = , = , = 0/ − Λ / . (17) The coupled system has the two eigenfrequencies with = : Sensors 2021, 21, 2533 8 of 15 1 = 1+ Λ − 4 + (Λ −1) (18) 2 r ! 1  2 2 =1 2 + 2 + + 2 − ω H= ω 1+1 1 Λ Λ+ 44 +κ (Λ Λ−1)1 (19)(19) 2 2 TheThe results results are are given given in in Figure Figure 44.. As expected, for very small κ,, the the lower lower (L) (L) and and higherhigher (H) (H) system system eigenfrequencies eigenfrequencies ( (ωL andand ωH,, respectively) crossed when ΛΛ=1= 1.. For For thisthis ,κ, both frequencies behaved as as uncoupled. uncoupled. The The larger larger κ got, the more the coupling coupling affectedaffected the the results, and onlyonly forfor largelarge mismatching,mismatching, the the resulting resulting frequencies frequencies were were close close to tothose those of of the the uncoupled uncoupled case. case.

Figure 4. Parameter dependence of the system eigenfrequencies and for a variation of the frequency-mistuning Figure 4. Parameter dependence of the system eigenfrequencies ω and ω for a variation of the frequency-mistuning parameter Λ. The zoom-level is increased from the left to the right inL the plot.H The used parameters were: m = 1 kg, k = 1 N/m,parameter and =10Λ. The zoom-level…10. is increased from the left to the right in the plot. The used parameters were: m = 1 kg, k = 1 N/m, and κ = 10−6 . . . 10−1. Interestingly, the more we zoomed in (figures from left to right), especially for Interestingly, the more we zoomed in (figures from left to right), especially for smaller smaller , a coupled region appeared in approximately 0.99 < Λ > 1.01, which showed κ, a coupled region appeared in approximately 0.99< Λ >1.01, which showed that, even that, even then, the co-resonance concept might be useful. However, the matching re- then, the co-resonance concept might be useful. However, the matching required extreme quired extreme precision: for =10, and should have deviated less than 1%. precision: for κ = 10−4, ω and ω should have deviated less than ±1%. Consequently, the Consequently, the larger resonator1 2 should be as small as possible. larger resonator should be as small as possible.

3.1.2.3.1.2. Frequency Response TheThe harmonic harmonic frequency frequency response response was was easi easilyly computed computed with with Equations Equations (14)–(16). (14)–(16). FigureFigure 5 shows an exemplaryexemplary result. This case was for =10κ = 10−4,, where where the the two two system system resonancesresonances ωL andand ωH were clearly visible in all three curves, and in GiU and GyU,, an an additionaladditional anti-resonance anti-resonance ωA appearedappeared at at ω1,, which which is is the the optimal vibration vibration migration migration frequencyfrequency for for the the application application of of a a tuned tuned mass mass damper. damper. Remarkably, Remarkably, there there was was no no further further antiresonanceantiresonance frequency frequency in in GiU thatthat could could be be expected, expected, possibly possibly due due to to the the already already high high transmissiontransmission factor factor =0.1α = 0.1. .This This is is promising promising because because the the corresponding corresponding phase phase zero zero crossingcrossing can can potentially potentially be be used used to to track track ωA with a phase feedback control. This This is is also also Sensors 2021, 21, x FOR PEER REVIEWthethe case case for for the the two two phase phase zero zero crossings crossings at at ωL and ωH.. For For these these two two frequencies, frequencies,9 ofself- self- 15

oscillationoscillation circuits circuits can can be be also also applied applied to to operate operate the the system system at at resonance. resonance.

Figure 5. Frequency response for , , and in magnitude (left) and phase (right). The used parameters were m = Figure 5. Frequency response for GiU, GyU, and GxU in magnitude (left) and phase (right). The used parameters were 4 5 −4 1 kg, mn = 0, k = 10 N/m, M = 10 kg, kM4 = 10 N/m, D5 = dM = 5x10 Ns/m, Cp −= 410 nF, and α = 0.1 N/V. m = 1 kg, mn = 0, k = 10 N/m, M = 10 kg, kM = 10 N/m, D = dM = 5 × 10 Ns/m, Cp = 10 nF, and α = 0.1 N/V. A closer look at the impact of the mass ratio in Figure 6 shows that the difference between and decreased for decreasing values. The main issue with a small fre- quency difference is that the coupling range becomes smaller and the system is more af- fected by production tolerances. Furthermore, the magnitude of increased with de- creasing values, which led to an elimination of the phase zero crossing in at . If became smaller than 10 , the two peaks of and joined, and both resonators decoupled. This is shown in Figure 6 on the right side. For a value so small, the fre- quency response resembled the response of a single piezoelectric element. However, the result was very promising for the co-resonance concept, since even for extremely small mass ratios (but still above =10), coupling is possible. Nevertheless, for smaller values, the initial tuning of and must be more accurate.

Figure 6. Magnitude of the admittance () for varying values, (left) 3D representation for a large range of , (right) 2D plot for very small mass ratios, and =10 …10 . The used parameters were m = 1 kg; mn = 0; k = 10 N/m; M = /; kM = /; D = dM = 5x10−4 Ns/m; Cp = 10 nF; and α = 0.1 N/V.

It is desirable for the operation of a nano-mass detector to have a phase zero crossing of at . The transmission factor was found being the main parameter in this study. Figure 7 shows its impact on the magnitude of for a constant =10 . For a large value, the antiresonance was clearly at ; with a decreasing value of , the moved toward . The piezoelectric coupling was not sufficient anymore to track the me- chanical antiresonance. The transmission factor, however, was not the only reason for the shifting . When the term j in Equation (14) became dominant, the other summand Sensors 2021, 21, x FOR PEER REVIEW 9 of 15

Figure 5. Frequency response for , , and in magnitude (left) and phase (right). The used parameters were m = 1 kg, mn = 0, k = 10 N/m, M = 104 kg, kM = 105 N/m, D = dM = 5x10−4 Ns/m, Cp = 10 nF, and α = 0.1 N/V. Sensors 2021, 21, 2533 9 of 15 A closer look at the impact of the mass ratio in Figure 6 shows that the difference between and decreased for decreasing values. The main issue with a small fre- quency differenceA closer look is that at the the impact coupling of the mass range ratio becomesκ in Figure smaller6 shows thatand thethe difference system is more af- between ω and ω decreased for decreasing κ values. The main issue with a small fected by productionL tolerances.H Furthermore, the magnitude of increased with de- frequency difference is that the coupling range becomes smaller and the system is more creasingaffected values, by production which tolerances.led to an elimination Furthermore, theof the magnitude phase ofzeroG crossingincreased in with at . If iU becamedecreasing smallerκ values, than which10 led, the to antwo elimination peaks of of the phaseand zero crossingjoined, in andGiU atbothωA. resonators −8 decoupled.If κ. became This smalleris shown than in10 Figure, the two 6 peakson the of rightωL and side.ωH joined, For a and bothvalue resonators so small, the fre- quencydecoupled. response This resembled is shown in the Figure response6 on the right of a side. single For a piezoelectricκ value so small, element. the frequency However, the response resembled the response of a single piezoelectric element. However, the result was result wasvery promisingvery promising for the co-resonance for the co-resonance concept, since evenconcept, for extremely since smalleven massfor extremely ratios small mass ratiosκ (but still (but above stillκ = above10−7), coupling=10 is possible.), coupling Nevertheless, is possible. for smaller Nevertheless,κ values, the for smaller initial tuning of ω and ω must be more accurate. values, the initial tuning1 of2 and must be more accurate.

Figure 6. Magnitude of the admittance ( ) for varying values, (left) 3D representation for a large range of , (right) Figure 6. Magnitude of the admittance (GiU) for varying κ values, (left) 3D representation for a large range of κ, (right) 2D −8 −5 2D plot for veryplot forsmall very mass small massratios, ratios, and and =10κ = 10 …10. . . 10 .. The The used used parameters parameters were mwere = 1 kg; m m=n 1= kg; 0; k m = 10n = N/m;0; k = M 10 = mN/m;/κ; M = /; −4 −4 kM = /; D =k Md=M k=/ κ5x10; D = d MNs/m;= 5 × 10Cp =Ns/m; 10 nF; C pand= 10 α nF; = and0.1 N/V.α = 0.1 N/V.

It is desirable for the operation of a nano-mass detector to have a phase zero crossing It is desirable for the operation of a nano-mass detector to have a phase zero crossing of GiU at ωA. The transmission factor α was found being the main parameter in this study. −4 of Figureat 7 shows. The itstransmission impact on the magnitudefactor ofwasGiU foundfor a constant beingκ =the10 main. For aparameter large α. in this study. value,Figure the 7 antiresonanceshows its impact was clearly on attheω 1magnitude; with a decreasing of value for of a αconstant, the ωA moved =10 . For a large toward value,ωL .the The antiresonance piezoelectric coupling was wasclearly not sufficient at ; anymorewith a decreasing to track the mechanical value of , the antiresonance. The transmission factor, however, was not the only reason for the shifting movedω towardA. When the term. ThejΩ piezoelectricCp in Equation (14)coupling became was dominant, not sufficient the other summand anymore with to the track the me- chanicalquadratic antiresonance. factor α significantly The transmission decreased. Eventually, factor, however, jΩCp limited was this not decrease. the only For ourreason for the application, we consequently looked for a large α while keeping C within limits. shifting . When the term j in Equation (14) became dominant,p the other summand Sensors 2021, 21, x FOR PEER REVIEW 10 of 15 Sensors 2021, 21, x FOR PEER REVIEW 10 of 15

with the quadratic factor significantly decreased. Eventually, j limited this de- Sensors 2021, 21, 2533 with the quadratic factor significantly decreased. Eventually, j limited this10 of de- 15 crease. For our application, we consequently looked for a large while keeping crease. For our application, we consequently looked for a large while keeping within limits. within limits.

FigureFigure 7. 7. MagnitudeMagnitude of the admittance admittance ( (G) for) for varying varying α values.values. The The used used parameters parameters were were m = Figure 7. Magnitude of the admittance ()iU for varying values. The used parameters were m = 1 kg; mn = 0; k = 10 N/m; M = /; kM = /; =10−4; D =− d4 M = 5 x 10−4 Ns/m;− C4p = 10 nF; and α = 0.1 m1 =kg; 1 kgmn; = m 0;n =k 0;= 10 k =N/m; 10 N/m; M = / M =;m k/Mκ =; k/M =; k/=10κ; κ−4;= D 10 = dM; D= 5 = x d 10M −=4 Ns/m; 5 × 10 Cp =Ns/m; 10 nF;C andp = α 10 =nF; 0.1 N/V. andN/V.α = 0.1 N/V. 3.1.3.3.1.3. Sensitivity Sensitivity on on Analyte Analyte Mass Mass 3.1.3. Sensitivity on Analyte Mass TheThe most most importantimportant factor of of a a system system is isth thee frequency frequency sensitivity sensitivity to the to analyte the analyte mass. The most important factor of a system is the frequency sensitivity to the analyte mass. mass.First, First,the corresponding the corresponding frequency frequency response response can be canobserved be observed in Figure in Figure8. Here,8. Here, denotesγ First, the corresponding frequency response can be observed in Figure 8. Here, denotes denotes / m and/ manand increasing an increasing correspondsγ corresponds to an toincreasing an increasing analyte analyte mass. mass. There There was no was ob- / andn an increasing corresponds to an increasing analyte mass. There was no ob- novious obvious change change in the in response the response for small for small . Thisγ. This was was in agreement in agreement with with the the findings findings in in vious change in the response for small . This was in agreement with the findings in FigureFigure2. 2. Figure 2.

Figure 8. Magnitude of the admittance () for varying values. The used parameters were m = Figure 8. Magnitude of the admittance () for varying values. The used parameters were m = Figure1 kg; m 8.n =Magnitude ; k = 10 N/m; of the M admittance = /; kM (=G /iU); for varying= 10−4; D γ= dvalues.M = 5 × 10 The−4 Ns/m; used C parametersp = 10 nF; and were 1 kg; mn = ; k = 10 N/m; M = /; kM = /; = 10−4; D− 4= dM = 5 × 10−4 Ns/m;−4 Cp = 10 nF; and mα = = 1 0.1 kg ;N/V. mn = γm; k = 10 N/m; M = m/κ; kM = k/κ; κ = 10 ; D = dM = 5 × 10 Ns/m; Cp = 10 nF; α = 0.1 N/V. and α = 0.1 N/V. Second, in order to visualize the frequency shift, we tracked three frequencies—both Second, in order to visualize the frequency shift, we tracked three frequencies—both resonanceSecond, frequencies in order to and visualize the antiresonance. the frequency Ma shift,ss detection we tracked is threebased frequencies—both on finding the zero resonance frequencies and the antiresonance. Mass detection is based on finding the zero resonancecrossings frequencies in the phase and of the. antiresonance.These frequencies Mass could detection be tracked is based by a on PLL finding (phase-locked- the zero crossings in the phase of . These frequencies could be tracked by a PLL (phase-locked- crossingsloop) control in the in phase a future of G device.iU. These With frequencies this data, could we computed be tracked the by relative a PLL (phase-locked- frequency shift loop)loop) control control in in a a future future device. device. With With this this data, data, we we computed computed the the relative relative frequency frequency shift shiftβ again (Figure9). Interestingly, the shift of the antiresonance frequency was twice as large as the shift of the resonance frequencies, where κ was again 10−4. Computations showed that this was independent of κ in the range of 10−5 < κ < 10−2 (data not shown). Compared Sensors 2021, 21, x FOR PEER REVIEW 11 of 15

Sensors 2021, 21, 2533 again (Figure 9). Interestingly, the shift of the antiresonance frequency was twice11 of 15as large as the shift of the resonance frequencies, where was again 10. Computations showed that this was independent of in the range of 10 <<10 (data not shown). Compared to the results in Figure 2, the sensitivity was about one order of mag- tonitude the results lower indue Figure to the2, theco-resonance sensitivity struct wasure, about but one detection order of and magnitude sensing lowerwere easier due tocompared the co-resonance to a sole nano-resonator. structure, but detection Moreover and, the sensing piezoelectric were easier approach compared further to simpli- a sole nano-resonator.fied the system. Moreover, In the future, thepiezoelectric the dimensions approach of the furthernano-resonator simplified could the system. be further In the re- future,duced. theThis dimensions is not possible of the for nano-resonator a sole nano-res couldonator be due further to issues reduced. related This to is the not actuation possible forand a measurement sole nano-resonator of the vibration. due to issues Therefore, related for to co-resonant the actuation systems, and measurement a higher sensitivity of the vibration.seems to be Therefore, within reach. for co-resonant systems, a higher sensitivity seems to be within reach.

FigureFigure 9.9. RelativeRelative frequency frequency changes changes (− (−)β due) due to the to the mass mass ratio ratio () for (γ) the for equivalent the equivalent model. model. The used parameters were m = 1 kg; mn= ; k = 10 N/m; M = /; kM = /; = 10−4; The used parameters were m = 1 kg; m = γ m; k = 10 N/m; M = m/κ; k = k/κ; κ = 10−4; −4 n M D = dM = 5 × 10 − Ns/m;4 Cp = 10 nF; and α = 0.1 N/V. D = dM = 5 × 10 Ns/m; Cp = 10 nF; and α = 0.1 N/V. 3.2.3.2. DeviceDevice DesignDesign WithWith thethe knowledgeknowledge fromfrom thethe equivalentequivalent model,model,an animportant importantstep steptowards towards a aspecific specific geometrygeometry couldcould nownow bebe made.made. Here,Here, wewe alsoalso consideredconsidered thethe fabricationfabrication processprocess toto ensureensure thatthat thethe designdesign isis suitedsuited forfor efficientefficient batchbatch production.production. InIn contrastcontrast toto previousprevious investiga-investiga- tionstions onon co-resonantco-resonant systems,systems, wewe proposepropose aa combinationcombination ofof aa longitudinallongitudinal andand aa bendingbending resonator.resonator. This concept concept offers offers several several advantages. advantages. In particular, In particular, the thefabrication fabrication of the of elec- the electrodestrodes for forthethe piezoelectric piezoelectric longitudinal longitudinal transducer transducer is less is less complicated. complicated. However, However, there there is isa adrawback: drawback: the the vibration vibration coupling coupling of of the the longit longitudinaludinal resonator to the substratesubstrate hashas toto bebe consideredconsidered in in the the design design process. process. Our Our fundamental fundamental setup setup is shown is shown in Figure in Figure1 . We 1. selected We se- AlNlected as AlN a piezoelectric as a piezoelectric material duematerial to its due good to acoustic its good properties acoustic andproperties its well-established and its well- fabricationestablished process. fabrication The process. nano-beam The wasnano-b madeeam of was gold made (Au) andof gold was (Au) supported and was by twosup- siliconported dioxideby two elements.silicon dioxide The piezoelectricelements. The element piezoelectric was 1 ×element1 × 1.4 wasµm 1 in x size.1 x 1.4 The µm Au in bottomsize. The electrode Au bottom was electrode buried by was the AlN.buried The by top the electrode AlN. The was top arranged electrode perpendicular was arranged toperpendicular the bottom electrode to the bottom and covered electrode the and top co andvered the the two top sides and of the the two AlN sides element. of the TheAlN nano-beamelement. The had nano-beam a cross-section had a cross-section of 320 × 210 of nm 320 and × 210 a free nm length and a free of 250 length nm. of The 250 SiO nm.2 × supportThe SiO elements2 support hadelements a height had of a 300height nm of and 300 a nm footprint and a offootprint 750 200 of 750 nm. × 200 nm. TheThe wholewhole systemsystem waswas modeledmodeled inin ANSYSANSYS Workbench,Workbench, wherewhere thethe piezoelectricpiezoelectric effecteffect waswas consideredconsidered byby usingusing thethe elementelement solid226.solid226. TheThe modemode shapesshapes werewere computedcomputed usingusing modalmodal analysis.analysis. AnAn independentindependent designdesign ofof thethe resonatorresonator dimensionsdimensions couldcould notnot bebe donedone sincesince thethe effectiveeffective massmass ofof bothboth resonatorsresonators shouldshould notnot havehave extremelyextremely differed.differed. Further-Further- more,more, thethe boundaryboundary conditionsconditions ofof thethe nano-beamnano-beam significantlysignificantly dependeddepended onon thethe elasticelastic properties of the longitudinal resonator. For the design of the longitudinal resonator, all properties of the longitudinal resonator. For the design of the longitudinal resonator, all relevant components needed to be considered in the model, including the ground plane relevant components needed to be considered in the model, including the ground plane and the supporting elements of the nano-beam. Consequently, the components had to be and the supporting elements of the nano-beam. Consequently, the components had to be simultaneously designed to obtain a suited setup. The selected dimensions were a result of simultaneously designed to obtain a suited setup. The selected dimensions were a result an empirical parameter study. For example, Figure 10 depicts the resonance frequencies of an empirical parameter study. For example, Figure 10 depicts the resonance frequencies as a function of the length (direction of vibration) of the piezoelectric AlN longitudinal as a function of the length (direction of vibration) of the piezoelectric AlN longitudinal resonator. Each marker indicates an eigenfrequency of the system. The result of the modal resonator. Each marker indicates an eigenfrequency of the system. The result of the modal analysis of the 3D FE Model did included not only the two coupled eigenmodes ω and ω analysis of the 3D FE Model did included not only the two coupled eigenmodesL andH but also all other modes in the investigated frequency range. Due to the geometry change, the order of the eigenmodes varied. The coupled modes were identified in the results and are connected by the blue solid lines. Again, the mode coupling was clearly visible, especially when compared to Figure4. To check if the other modes (markers without line) Sensors 2021, 21, x FOR PEER REVIEW 12 of 15

but also all other modes in the investigated frequency range. Due to the geometry Sensors 2021, 21, 2533 change, the order of the eigenmodes varied. The coupled modes were identified 12in ofthe 15 results and are connected by the blue solid lines. Again, the mode coupling was clearly visible, especially when compared to Figure 4. To check if the other modes (markers with- out line) in the interesting frequency range were also interacting with the two coupled eigenmodes,in the interesting harmonic frequency simulations range were (where also the interacting piezo was with driven the two electrically) coupled eigenmodes, will be car- riedharmonic out next simulations to the modal (where analysis. the piezo Those was drivenresults electrically)in Figure 12 will show be carriedthat only out the next two to the modal analysis. Those results in Figure 12 show that only the two coupled longitudinal- coupled longitudinal-bending modes were excitable. In the same way, additional geomet- bending modes were excitable. In the same way, additional geometrical parameters were rical parameters were simulated. To obtain the best configuration, a multi parameter op- simulated. To obtain the best configuration, a multi parameter optimization is needed in timization is needed in the future. the future.

Figure 10. Parameter dependence of all eigenfrequencies on the different length of the piezoelec- Figure 10. Parameter dependence of all eigenfrequencies on the different length of the piezoelectric tric element in the frequency range from 450 to 900 MHz, as indicated by makers. The lines show element in the frequency range from 450 to 900 MHz, as indicated by makers. The lines show the the change of the two utilized eigenmodes and . change of the two utilized eigenmodes ωL and ωH.

The frequency shift, due to the analyte mass, of the two coupled modes have been computed by modal analysis. The The results results are are shown shown in in Figure Figure 1111 withwith unloaded unloaded frequen- frequen- cies of 666 and 697 MHz. The frequency shift is determined by ( −,)/(2) and cies of 666 and 697 MHz. The frequency shift is determined by (ωL − ωL,0)/(2π) and ( −,)/(2), respectively, where the index “0” indicates the unloaded case. The un- (ωH − ωH,0)/(2π), respectively, where the index “0” indicates the unloaded case. The coupleduncoupled frequencies frequencies could could not not be be determined determined since since the the boundary boundary conditions conditions were were too complex. To estimate thethe sensitivitysensitivity toto thethe analyteanalyte mass, mass, a a small small element element (25 (25× × 2525 ×× 2525 nm) nm) Sensors 2021, 21, x FOR PEER REVIEWwas attached to to the the top top center center of of the the nano-b nano-beam.eam. The The weight weight was was adapted adapted by bychanging changing13 ofits 15

density.its density. The sensitivity was approximately 250 Hz/ag. Due to the more complex geometry compared to the equivalent model, the sensitivity of both modes was different. Conse- quently, we expected a detectable mass of 400 yg (4 10 g) if the measurement electron- ics had a 100 mHz resolution. According to Table 1, this was in the range of the carbon- nanotube based detectors, especially the detector of [12] showing an approximately three times better resolution. The advantage of the novel design is its simplicity in setup, that allows for a rather easy fabrication compared to the already known devices. For the de- tection of a single digit yoctogram, we aim for both further miniaturization and an im- provement in the frequency resolution of the measurement electronics towards a few mHz. The hitherto theoretical concept therefore has the potential to achieve a resolution in the range of the atomic mass unit (1.66 10 g) if both measures can be successfully implemented. However, according to the 3 definition of the limit of detection (LOD), detecting 1.66 10 g does not only require a frequency resolution of the measurement electronics

ofFigure a few 11.mHz Absolute for improved frequency sensitivity change of (but −also, a)/(2) frequency and (noise − of, the)/(2) zero duesignal to an of ana- Figure 11. Absolute frequency change of (ωL − ωL,0)/(2π) and (ωH − ωH,0)/(2π) due to an analyte lyte = 1.66 mass 10 attached 1000 to 10the top center= 1.66mHz of the nano-beamat a given and sensitivity, corresponding e.g., coupled of 1000 eigenmodes. , which mass attached to the top center of the nano-beam and corresponding coupled eigenmodes. seems quite challenging. TheElectrical sensitivity behavior was approximatelyis crucial for the 250 sensin Hz/ag.g process; Due totherefore, the more a harmonic complex geometrysimulation comparedwas computed. to the The equivalent system was model, driven the with sensitivity a 1 V amplitude, of both modes and the was corresponding different. Con- fre- − sequently,quency response we expected is a shown detectable in Figure mass 12. of It 400 was yg evident (4 × 10 that22 g) the if capacity the measurement dominated electronics(compare hadFigure a 100 7), mHzhence resolution. two clear Accordingresonance–anti-resonance to Table1, this was combinations in the range appeared. of the carbon-nanotubeBoth had a phase based zero detectors,crossing, especiallyand the minimum the detector phase of [was12] showingabout −60°, an approximatelywhich was suf- ficient for both PLL tracking and a self-oscillation approach. The dominant capacity did not allow for the tracing of the = frequency by a PLL circuitry for this parameter combination. However, the simulated results underlined that the novel concept of com- bining a longitudinal and a bending resonator is feasible and very promising.

Figure 12. Current amplitude in the magnitude and phase of the co-resonant vibration system.

4. Conclusions With the presented theoretical investigation, the feasibility of the novel combination of longitudinal and bending modes for co-resonant nano-mass detectors has been demon- strated. The new design simplifies significantly fabrication compared to many state-of- the-art structures used in other approaches. The sensitivity was, at first sight, about just one order of magnitude smaller that in non-co-resonant systems. Nevertheless, due to the simple design of the nano-beam, it has potential for its size to be further reduced. The direct application of the piezoelectric element for actuation and sensing appears very promising and also simplifies the system. The detection of the mechanical anti-resonance in through the phase zero crossing of seems to be possible but very challenging. This has the potential of an additional factor of 2 in the resolution. These results motivate the next steps: to prove the model-based findings in the labor- atory. Here, we will work on the presented device with an expected resolution of 400 yg. Sensors 2021, 21, x FOR PEER REVIEW 13 of 15

Sensors 2021, 21, 2533 13 of 15

three times better resolution. The advantage of the novel design is its simplicity in setup, that allows for a rather easy fabrication compared to the already known devices. For the detection of a single digit yoctogram, we aim for both further miniaturization and an improvement in the frequency resolution of the measurement electronics towards a few mHz. The hitherto theoretical concept therefore has the potential to achieve a res- olution in the range of the atomic mass unit (1.66 × 10−24 g) if both measures can be successfully implemented. However, according to the 3σ definition of the limit of detection (LOD), detecting 1.66 × 10−24. g does not only require a frequency resolution of the measurement electronics ofFigure a few 11. mHz Absolute for improvedfrequency change sensitivity of ( but− also,)/(2) a frequency and ( noise−,)/(2) of the zerodue to signal an ana- of lyte mass attached−24 to the top center18 Hz of the nano-beam and corresponding coupled eigenmodes.Hz σ = 1.66 × 10 × 1000 × 10 g = 1.66mHz at a given sensitivity, e.g., of 1000 ag , which seems quite challenging. ElectricalElectrical behaviorbehavior isis crucialcrucial forfor thethe sensingsensing process;process; therefore,therefore, aa harmonicharmonic simulationsimulation waswas computed.computed. The The system system was was driven driven with with a 1 a V 1 Vamplitude, amplitude, and and the thecorresponding corresponding fre- quency response is shown in Figure 12. It was evident that the capacity dominated frequency response GiU is shown in Figure 12. It was evident that the capacity dominated (compare(compare FigureFigure7 ),7), hence hence two two clear clear resonance–anti-resonance resonance–anti-resonance combinations combinations appeared. appeared. BothBoth had a phase zero zero crossing, crossing, and and the the minimum minimum phase phase was was about about −60°,−60 which◦, which was was suf- sufficientficient for for both both PLL PLL tracking tracking and and a self-oscillation a self-oscillation approach. approach. The The dominant dominant capacity capacity did didnot notallow allow for forthe thetracing tracing of the of the ω=1 = ω frequencyA frequency by bya PLL a PLL circuitry circuitry for forthis this parameter param- etercombination. combination. However, However, the simulated the simulated result resultss underlined underlined that the that novel the novel concept concept of com- of combiningbining a longitudinal a longitudinal and anda bending a bending resonator resonator is feasible is feasible and and very very promising. promising.

FigureFigure 12.12. CurrentCurrent amplitudeamplitude inin thethe magnitudemagnitude andand phasephase ofof thethe co-resonantco-resonant vibrationvibration system.system.

4.4. ConclusionsConclusions WithWith thethe presented theoretical theoretical investigation, investigation, the the feasibility feasibility of ofthe the novel novel combination combina- tionof longitudinal of longitudinal and bending and bending modes modes for co-r foresonant co-resonant nano-mass nano-mass detectors detectors has been has demon- been demonstrated.strated. The new The design new simplifies design simplifies significant significantlyly fabrication fabrication compared compared to many to state-of- many state-of-the-artthe-art structures structures used in used other in approaches. other approaches. The sensitivity The sensitivity was, was,at first at firstsight, sight, about about just justone oneorder order of magnitude of magnitude smaller smaller that thatin non-co in non-co-resonant-resonant systems. systems. Nevertheless, Nevertheless, due to due the tosimple the simple design design of the ofnano-beam, the nano-beam, it has itpotential has potential for its for size its to size be to further be further reduced. reduced. The Thedirect direct application application of the of the piezoelectric piezoelectric elem elementent for for actuation actuation and and sensing sensing appears veryvery promisingpromising andand alsoalso simplifiessimplifies thethe system.system. TheThe detectiondetection ofof thethe mechanicalmechanical anti-resonanceanti-resonance in G through the phase zero crossing of G seems to be possible but very challenging. in yU through the phase zero crossing of iU seems to be possible but very challenging. ThisThis hashas thethe potentialpotential ofof anan additionaladditional factorfactor ofof 22 inin thethe resolution.resolution. TheseThese results motivate motivate the the next next steps: steps: to prove to prove the model-based the model-based findings findings in the in labor- the laboratory.atory. Here, Here, we will we work will workon the on presented the presented device device with an with expected an expected resolution resolution of 400 yg. of 400 yg. For this, the design will be further improved hand-in-hand with the technology development of the fabrication process. The novel device will then be experimentally characterized and evaluated. Sensors 2021, 21, 2533 14 of 15

Author Contributions: Conceptualization, J.T., A.G., M.C.W., M.S., M.H., and S.Z.; methodology, formal analysis, and investigation, J.T.; writing—original draft preparation, J.T.; writing—review and editing, J.T., M.C.W., and S.Z.; visualization, J.T. and S.d.W. All authors have read and agreed to the published version of the manuscript. Funding: The publication of this article was funded by the Open Access fund of Leibniz Univer- sität Hannover. Informed Consent Statement: Not applicable. Data Availability Statement: The data are available upon request. Acknowledgments: The authors are grateful for the publication funding by the Open Access fund of Leibniz Universität Hannover. Conflicts of Interest: The authors declare no conflict of interest.

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